13 o Dr. Robertson's expeditions methods of calculating 
€M -f- CS : CM— CS : : tan . £ ACM : tan. | (CSM — CMS). 
Hence the angles become known by their sum and difference. 
As the angle SMN is very small, and consequently the angle 
MSC nearly equal to EGA in the orbits of almost all the 
planets, this way of finding the angle CSM is usually called 
Cassini's approximation to the excentric anomaly ACE. 
First Method. 
Having found the angle CSM, sin. CSM : CM : : sin. SCM : 
SM, which therefore becomes known. Let % equal the angle 
SMN, 5 equal the series expressing its sine, and c equal the 
series expressing its cosine. Put a equal the sine of CMS, 
and b equal its cosine. Then, radius being i, ac — > bs = sin. 
CMN == sin. ECM = sin. (CMS — z). Also, i : SM : : s : 
SN = SM x s , and ac — • bs -f- SM x s = TN -f- SN = EM ==• 
CMS- — z, and therefore CMS = z -}- ac — bs + SM x s = z -f- ac 
S 1VX b x s. 
hetd=SM — b, and then CMS — z-^-ac-^ds—z-^-a (l — — 
+ — + £!___ &c.) + d!z-~ + — Sal.) 
* n n A -7 1 A C 1 2. 1.A . C.6.7.8 ' * V 2.2 * Z.1.A-.Z / 
2.3.4 2 3-4-5 -6 
! a — 1~ % dz 
2.34.5.6.7.8 
az 2, dz 3 , 
2 2.-2 l " 
azr 
+ 
dx s 
2.34.5 
— &C. 
2.3. 1 2,34 1 2.34.5 
Let g=CMS — a, and putting A, B, C, &c. for the coefficients 
e = Az — Bz°~ — Cz 3 + Bte 4 -j- E% 5 — F z 6 — Gz 7 -f- &c. 
1 . a 
By reversing this series we find % = 
d 
6(1 4-rf) 4 
r 3 4 - -7- 
8 2(1 
24(1 + d) 5 
1 + d 
$ad 
12(1 +d ) 6 
e -f- 
^ +. StT+dp 
2(14 -d ) 3 
5a 3 
+ &C. 
This equation is in parts of the radius, and in order to have 
it in degrees we use this proportion, 1:57°. 2957795 : :z; 57. 
S 95779S * — R° x, putting R° for 57°. 2957795- 
